A316184 - OEIS

A316184

Positive integers R such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.

1, 7, 9, 35, 37, 91, 183, 189, 341, 559, 845, 855

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..12.

Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars.

EXAMPLE

For the first entry of R=1, we have the polynomial x^3 - 3x + 1. Its roots, expressed as continued fractions, all have a common tail of 3, 2, 3, 1, 1, 6, 11, ... The next examples are R=7 with the polynomial x^3 - 7x + 7, then R=9 with the polynomial x^3 - 9x + 9, and Q=35 with the polynomial x^3 - 21x + 35. Note that for the R=7 example, we get the common tail of 2, 3, 1, 6, 10, 5, ... which is contained in A039921.

MATHEMATICA

SetOfQRs = {};

M = 1000;

Do[

If[Divisible[3 (a^2 - a + 1), c^2] &&

Divisible[(2 a - 1) (a^2 - a + 1), c^3] &&

3 (a^2 - a + 1)/c^2 <= M,

SetOfQRs =

Union[SetOfQRs, { { (3 (a^2 - a + 1))/

c^2, ((2 a - 1) (a^2 - a + 1))/c^3}} ]],

{c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}];

Print[SetOfQRs // MatrixForm];

CROSSREFS

Cf. A039921, A316157.

Sequence in context: A261961 A177030 A189974 * A359367 A321760 A083203

Adjacent sequences: A316181 A316182 A316183 * A316185 A316186 A316187

KEYWORD

nonn,more

AUTHOR

Greg Dresden, Jun 25 2018

STATUS

approved